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Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising

Consecutive Negative Numbers

Stage: 3 Challenge Level:

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Summing Consecutive Numbers

Stage: 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Cut it Out

Stage: 2 Challenge Level:

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Have You Got It?

Stage: 3 Challenge Level:

Can you explain the strategy for winning this game with any target?

Walking the Squares

Stage: 2 Challenge Level:

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Number Pyramids

Stage: 3 Challenge Level:

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Steps to the Podium

Stage: 2 and 3 Challenge Level:

It starts quite simple but great opportunities for number discoveries and patterns!

Christmas Chocolates

Stage: 3 Challenge Level:

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Card Trick 2

Stage: 3 Challenge Level:

Can you explain how this card trick works?

Maths Trails

Stage: 2 and 3

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

More Magic Potting Sheds

Stage: 3 Challenge Level:

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Nim-7

Stage: 1, 2 and 3 Challenge Level:

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Triangle Numbers

Stage: 3 Challenge Level:

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

Make 37

Stage: 2 and 3 Challenge Level:

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Partitioning Revisited

Stage: 3 Challenge Level:

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

More Number Pyramids

Stage: 3 Challenge Level:

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Picturing Triangular Numbers

Stage: 3 Challenge Level:

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Enclosing Squares

Stage: 3 Challenge Level:

Can you find sets of sloping lines that enclose a square?

Spirals, Spirals

Stage: 2 Challenge Level:

Here are two kinds of spirals for you to explore. What do you notice?

Mind Reading

Stage: 3 Challenge Level:

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

Hidden Rectangles

Stage: 3 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Fault-free Rectangles

Stage: 2 Challenge Level:

Find out what a "fault-free" rectangle is and try to make some of your own.

Handshakes

Stage: 3 Challenge Level:

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Cunning Card Trick

Stage: 3 Challenge Level:

Delight your friends with this cunning trick! Can you explain how it works?

Play to 37

Stage: 2 Challenge Level:

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Nim-7 for Two

Stage: 1 and 2 Challenge Level:

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Squares in Rectangles

Stage: 3 Challenge Level:

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Snake Coils

Stage: 2 Challenge Level:

This challenge asks you to imagine a snake coiling on itself.

One O Five

Stage: 3 Challenge Level:

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Round the Four Dice

Stage: 2 Challenge Level:

This activity involves rounding four-digit numbers to the nearest thousand.

Centred Squares

Stage: 2 Challenge Level:

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Konigsberg Plus

Stage: 3 Challenge Level:

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Strike it Out for Two

Stage: 1 and 2 Challenge Level:

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

Always, Sometimes or Never?

Stage: 1 and 2 Challenge Level:

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Got it for Two

Stage: 2 Challenge Level:

Got It game for an adult and child. How can you play so that you know you will always win?

Build it Up

Stage: 2 Challenge Level:

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

Tiling

Stage: 2 Challenge Level:

An investigation that gives you the opportunity to make and justify predictions.

Roll over the Dice

Stage: 2 Challenge Level:

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

Tourism

Stage: 3 Challenge Level:

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Triangle Pin-down

Stage: 2 Challenge Level:

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Cubes Within Cubes Revisited

Stage: 3 Challenge Level:

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

More Twisting and Turning

Stage: 3 Challenge Level:

It would be nice to have a strategy for disentangling any tangled ropes...

All Tangled Up

Stage: 3 Challenge Level:

Can you tangle yourself up and reach any fraction?

Got It

Stage: 2 and 3 Challenge Level:

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Build it up More

Stage: 2 Challenge Level:

This task follows on from Build it Up and takes the ideas into three dimensions!

Number Differences

Stage: 2 Challenge Level:

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Dice Stairs

Stage: 2 Challenge Level:

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Domino Numbers

Stage: 2 Challenge Level:

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Magic Vs

Stage: 2 Challenge Level:

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Tilted Squares

Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?